- past of triangulate
In trigonometry and geometry, triangulation is the process of finding coordinates and distance to a point by calculating the length of one side of a triangle, given measurements of angles and sides of the triangle formed by that point and two other known reference points, using the law of sines.
In the figure at right, the third angle of the triangle (call it θ) is known to be 180 − α − β, since the sum of the three angles in any triangle is known to be 180 degrees. The opposite-side for this (the third) angle is l, which is a known distance. Since, by the law of sines, the ratio sin(θ)/l is equal to that same ratio for the other two angles α and β, the lengths of any of the remaining two sides can be computed by algebra. Given either of these lengths, sine and cosine can be used to calculate the offsets in both the north/south and east/west axes from the corresponding observation point to the unknown point, thereby giving its final coordinates.
Some identities often used (valid only in flat or euclidean geometry):
- α, β and distance AB are already known
- C can be calculated by using the distance RC or MC:
- RC: Position of C can be calculated using the Law of Sines
Now we can calculate AC and BC
Last step is to calculate RC via
- RC=AC \cdot \sin\alpha
- RC=BC \cdot \sin\beta
- MC can be calculated using the Law of Cosines and the Pythagorean theorem
- MR=AM-RB=\left(\frac\right)-\left(BC \cdot \cos\beta\right)
Triangulation is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and gun direction of weapons.
Many of these surveying problems involve the solution of large meshes of triangles, with hundreds or even thousands of observations. Complex triangulation problems involving real-world observations with errors require the solution of large systems of simultaneous equations to generate solutions.
Famous uses of triangulation have included the retriangulation of Great Britain.
triangulated in Bulgarian: Триангулация
triangulated in Czech: Triangulace
triangulated in German: Triangulation (Geodäsie)
triangulated in Spanish: Triangulación
triangulated in French: Triangulation
triangulated in Croatian: Triangulacija
triangulated in Indonesian: Triangulasi
triangulated in Italian: Triangolazione
triangulated in Hebrew: טריאנגולציה
triangulated in Hungarian: Háromszögelés
triangulated in Dutch: Driehoeksmeting
triangulated in Norwegian: Triangulering
triangulated in Polish: Triangulacja (geodezja)
triangulated in Russian: Триангуляция
triangulated in Slovenian: Triangulacija
triangulated in Swedish: Triangulering